Let $\mathcal{M}$ be a von Neumann algebra without direct commutative summands, and let $\mathcal{A}$ be an arbitrary subalgebra of $LS(\mathcal{M})$ containing $\mathcal{M},$ where $LS(\mathcal{M})$ is the ${}^{\ast }$-algebra of all locally measurable operators with respect to $\mathcal{M}$. Suppose $\delta$ is an additive mapping from $\mathcal{A}$ to $LS(\mathcal{M})$ that satisfies the condition $\delta (A)B^{\ast }+A\delta (B)+\delta (B)A^{\ast }+B\delta (A)=0$ whenever $AB=BA=0.$ In this paper, we prove that there exists an element $Y$ in $LS(\mathcal{M})$ such that $\delta (X)=XY-Y{X}^{\ast },$ for every $X$ in $\mathcal{A}.$